The uncertainty that arises in the EiV model from estimating \(\zeta \)  given \(x\)  is an aleatoric uncertainty that has to be taken into account whenever predicting \(f_\theta (\zeta )\) . It explains why there can be an uncertainty of the prediction even if there is no epistemic uncertainty left, without forcing this role on the output noise.
We will approach (REF ) from a Bayesian point of view which is a consistent way for describing uncertainty but poses the challenge to sample from a high dimensional posterior on \(\theta \) . To overcome this obstacle, we build on the idea to use variational inference, as e.g. in [1]}, [2]}, [3]}, [4]}. In fact our approach is agnostic to the specific form of the variational distribution and could be used to extend any of these works. For convenience the results in Section  below were produced with the variational distribution induced by Monte Carlo dropout [2]}, but let us stress that there is no profound reason for using this approach in this work other than its ease of implementation. In fact, studying how well different methods perform under EiV could be an exciting outlook for future work.
